How to define tensor contraction without referring to summation? The textbook defines a tensor to be an element in $(T^*)^k×T^l→R$. It then expresses tensors as arrays of components with respect to a certain basis, and defines tensor contraction using summation convention. My question is: Can we define tensor contraction without referring to basis,  components, and summation, but just the definition of tensor as a multilinear function? If components are inescapable, why bother define tensors as multilinear functions?
 A: I can give you a mathematical reason why you have to use components to define contraction.
The reason is that contraction doesn't work for infinite dimensional vector spaces. You would have to sum over infinitely many components, and this sum might not converge. So there is no possibility of contraction for infinite dimensional tensors.*
But if we had a "basis free" definition of contraction then there would be nothing to stop it working in the infinite dimensional case as well. The definition of finite dimensional is "has a finite basis", and so the definition always has to mention a basis in order to distinguish between the finite and infinite dimensional cases.
So using a basis really is inescapable here.
(In the comments to your question ACuriousMind gives a component free definition of contraction, but this relies on a slightly different definition of tensor than the one you were given. I order to prove that the definitions are the same you again have to assume finite dimensionality and pick a basis, because the definitions aren't the same in the infinite dimensional case.)

If components are inescapable, why bother define tensors as multilinear functions?

In physics there's no "preferred basis". So an answer to a physical question can't be correct if it depends on which basis you pick. After defining contraction, your teacher should have shown that if you were to pick a different basis to write your coordinates in then you would indeed get the same answer as before.
Doing this checking-that-things-don't-depend-on-the-basis is annoying, and so it's often best to give basis free definitions. (Views differ on when one should use a basis. Some people work in a basis all the time; others avoid them like the plague.)
*Example: The tensor $\delta^a_{\;b}$ would have infinitely many ones on its diagonal, so $\delta^a_{\;a}$ wouldn't exist.
